On compact embbeded Weingarten hypersurfaces in warped products

DOI10.1016/J.JMAA.2022.126593arXiv2206.06727MaRDI QIDQ6402026

Julien Roth, Abhitosh Upadhyay

Publication date: 14 June 2022

Abstract: We show that compact embedded starshaped $r$-convex hypersurfaces of certain warped products satisfying $H_r=aH+b$ with $ageqslant0$, $b>0$, where $H$ and $H_r$ are respectively the mean curvature and $r$-th mean curvature is a slice. In the case of space forms, we show that without the assumption of starshapedness, such Weingarten hypersurfaces are geodesic spheres. Finally, we prove that, in the case of space forms, if $Hr-aH-b$ is close to $0$ then the hypersurface is close to geodesic sphere for the Hausdorff distance. We also prove an anisotropic version of this stability result in the Euclidean space.

Mathematics Subject Classification ID

Minimal surfaces in differential geometry, surfaces with prescribed mean curvature (53A10) Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) (53C42)

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